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An arithmetic count of the lines on a smooth cubic surface

Publication ,  Journal Article
Leo Kass, J; Wickelgren, K
Published in: Compositio Mathematica
April 1, 2021

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field k, generalizing the counts that over C there are 27 lines, and over R the number of hyperbolic lines minus the number of elliptic lines is 3. In general, the lines are defined over a field extension L and have an associated arithmetic type α in L∗/(L∗)2. There is an equality in the Grothendieck–Witt group GW(k) of k, ∑ TrL/k <α> = 15 · <1> + 12 · <−1>, lines where TrL/k denotes the trace GW(L) → GW(k). Taking the rank and signature recovers the results over C and R. To do this, we develop an elementary theory of the Euler number in A1-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

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Published In

Compositio Mathematica

DOI

ISSN

0010-437X

Publication Date

April 1, 2021

Volume

157

Issue

4

Start / End Page

677 / 709

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics
 

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Leo Kass, J., & Wickelgren, K. (2021). An arithmetic count of the lines on a smooth cubic surface. Compositio Mathematica, 157(4), 677–709. https://doi.org/10.1112/S0010437X20007691
Leo Kass, J., and K. Wickelgren. “An arithmetic count of the lines on a smooth cubic surface.” Compositio Mathematica 157, no. 4 (April 1, 2021): 677–709. https://doi.org/10.1112/S0010437X20007691.
Leo Kass J, Wickelgren K. An arithmetic count of the lines on a smooth cubic surface. Compositio Mathematica. 2021 Apr 1;157(4):677–709.
Leo Kass, J., and K. Wickelgren. “An arithmetic count of the lines on a smooth cubic surface.” Compositio Mathematica, vol. 157, no. 4, Apr. 2021, pp. 677–709. Scopus, doi:10.1112/S0010437X20007691.
Leo Kass J, Wickelgren K. An arithmetic count of the lines on a smooth cubic surface. Compositio Mathematica. 2021 Apr 1;157(4):677–709.
Journal cover image

Published In

Compositio Mathematica

DOI

ISSN

0010-437X

Publication Date

April 1, 2021

Volume

157

Issue

4

Start / End Page

677 / 709

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics